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E. T. S. I. Caminos, Canales y Puertos7 1) Requires two initial estimates: x i-1 and x i These do NOT have to bracket root ! 2) Maintains a strict sequence: Repeated until: a. | f(x i+1 ) | < k with k = small number b. c. Max. number of iterations is reached. 3. If x i and x i+1 were to bracket the root, this would be the same as the False-Position Method. BUT WE DON'T! Open Methods (Secant Method)

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E. T. S. I. Caminos, Canales y Puertos13 can be small, even though x new is not close to root. Remedy: Do not completely rely on  a to ensure that the problem is solved. Check to make sure |f(x new ) | < . Convergence Problem: For slowly converging functions Open Methods (Fixed point method)

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E. T. S. I. Caminos, Canales y Puertos16 Why do open methods fail? Function may not look linear. Remedy: recast into a linear form. For example, Is a poorly constrained problem in that there is a large, nearly flat zone for which the derivative is near zero. Recast as: i f(i) = 0 = 7,500 i [ 1 - (1+i) -20 ] Open Methods

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E. T. S. I. Caminos, Canales y Puertos21 A real rootfinding problem can be viewed as having three phases: 1) Opening moves: One needs to find the region of the parameter space in which desired root can be found. Understanding of problem, physical insight, and common sense are valuable. 2) Middle Game: Use robust algorithm to reduce initial region of uncertainty. 3) End game: Generate a highly accurate solution in a few iterations. Three Phase Rootfinding Strategy